Automatica 44 (2008) 472 – 478 www.elsevier.com/locate/automatica Brief paper Multivariable PID control with set-point weighting via BMI optimisation  Fernando D. Bianchi ∗, 1 , Ricardo J. Mantz 2 , Carlos F. Christiansen 3 Laboratorio de Electrónica Industrial, Control e Instrumentación (LEICI), Facultad de Ingeniería, Universidad Nacional de La Plata, CC 91, 1900 La Plata, Argentina Received 6 June 2006; received in revised form 28 December 2006; accepted 31 May 2007 Available online 14 September 2007 Abstract The paper focuses on the design of multivariable PID controllers with set-point weighting. The advantage of this PID structure is that the responses of the system to disturbances and to changes in the set-point can be adjusted separately. The proposed design methods rely on the transformation of the tuning of the controller gains into a static output feedback (SOF) problem. Hence, multivariable PID controllers can be designed by solving an optimisation problem with bilinear matrix inequalities (BMIs). The paper addresses the design of both time-invariant and gain-scheduled robust controllers. All of the tuning methods discussed through the paper are based on a PID structure with filtered derivative term, thus guaranteeing the well-posedness of the closed loop system.  2007 Elsevier Ltd. All rights reserved. Keywords: PID control; Robust control; Gain scheduling; Bilinear matrix inequality (BMI); Linear parameter varying systems 1. Introduction Although new and more powerful tools have been devel- oped, PID control is still the most used control strategy in in- dustrial applications. An attractive feature of PID controllers is their relatively simple and intuitive design. Moreover, the fixed structure of PID controllers has made possible the development of ready-made hardware modules and software packages for a quick and easy implementation (Li, Ang, & Chong, 2006). For these reasons, PID controllers are commonly preferred even though more aggressive controllers can be obtained with other more sophisticated techniques. The popularity of PID controllers has encouraged the formu- lation of a large number of methods for tuning the controller parameters (see e.g., Astrom & Hagglund, 2005; O’Dwyer,  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Mario Sznaier under the direction of Editor Roberto Tempo. ∗ Corresponding author. Tel./fax: +54 221 4259306. E-mail address: fbianchi@ing.unlp.edu.ar (F.D. Bianchi). 1 UNLP-CONICET. 2 UNLP-CICpBA. 3 UNLP. 0005-1098/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.05.021 2006). In MIMO plants with low interactions, it is possible to employ PID controllers in multiple SISO loops selected according to physical considerations or to a relative gain ar- ray (RGA) analysis. In this case, decentralised multivariable control, classical or indeed heuristic tuning methods can be used. However, in plants having strong interactions between the input and output pairs, it is necessary the use of centralised controllers designed with more sophisticated tuning methods usually based on optimisation notions (see e.g., Ruiz-Lopez, Rodriguez-Jimenes, & Garcia-Alvarado, 2006 and references therein). An approach, introduced in Ge, Chiu, and Wang (2002) and Zheng, Wang, and Lee (2002), consists in trans- forming the tuning of the controller parameters into a static output feedback (SOF) problem. Then, the controller parame- ters are determined by solving an optimisation problem with bilinear matrix inequalities (BMIs) for which several numer- ical algorithms are currently available (see e.g., Apkarian, Noll, & Tuan, 2003; El Ghaoui, Oustry, & AitRami, 1997; Goh, Safonov, & Papavassilopoulus, 1994; Orsi, Helmke, & Moore, 2006). A similar approach has also been used to formulate effective tuning methods for robust and gain- scheduled multivariable PID controllers (Mattei, 2001). It is interesting to note that these methods are actually particular cases of the procedures for designing optimal fixed-structure